Value of Cos 120

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How to Find Value of Cos 120 Degree?

Cos 120 degree is equal to minus half which can be written as (- ½ ) or (-0.5).  Cos is one of the functions of Trigonometry that deals with the relationship between the angles and sides of a right-angled triangle. So in short, we can say that measuring a triangle (specifically right-angled triangle) is trigonometry. 

 

Now, the question arises what is a right-angled Triangle? We know that a triangle is a closed figure with three sides and three angles. The triangle having one of its interior angles as 90 degrees and the other two angles less than 90 degrees is called a right-angled triangle. Some of the important functions of Trigonometry are Sin, Cos, and Tan. The use of these functions finds its applications in the field of astronomy, engineering, architectural design, and physics. 

 

Let us take a right-angled triangle, select one of the angles as () and name three sides as follow:

  • Hypotenuse: The longest side of the triangle which is opposite to 90 degrees.

  • Perpendicular (Opposite): It is the side opposite to the unknown angle and perpendicular to the base (that is, the angle between base and perpendicular is 90 degrees).

  • Base (Adjacent): It is the side on which triangle rests and it also contains both the angles (90 degrees and unknown angle .

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Most of the trigonometry calculations are done by using the trigonometry ratios. There are 6 trigonometry ratios present in trigonometry. Every other important trigonometry formula is derived with the help of these ratios. 

 

Given below are the important ratios named as Sin, Cos, and Tan. Sin and cos are the fundamental or basic ratios whereas Tan, sec, cot, and sec are derived functions. Given below are the ratios of Sin, Cos, and Tan.

(i) Sine Function (sin)

sin \[\theta\] = \[\frac{Opposite}{Hypotenuse}\]

 

(ii) Cosine Function (cos)

cos \[\theta\] = \[\frac{Adjacent}{Hypotenuse}\]

 

(iii) Tangent Function (tan)

tan (\[\theta\]) = \[\frac{sin (\theta) }{cos (\theta) }\]

 

Quadrant and Cast Rule

As stated above, the angles other than the 90 degrees angle in a right-angled triangle are acute (i.e, less than 90 degrees). To find the value of functions for angles more than 90 degrees, we follow a set of rules known as cast rule. Let us take four axes, to divide 360 degrees into four quadrants. The angles are always measured anti-clockwise from the positive x-axis. The given diagram shows 30 degrees from the positive x-axis.

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There are four quadrants as shown in the figure below. Each quadrant contains a range of angles:

 

First Quadrant: All the angles between 0 and 90 lie in the first quadrant. The value of all the functions (Sin, Tan, Cos) in this quadrant are positive. Shown as A (represents All) in the second diagram given below. 

 

Second Quadrant: All the angles between 90 and 180 lie in the second quadrant. The value of only Sin is positive in this quadrant. Shown as S (represents Sin) in the second diagram given below. 

 

Third Quadrant: All the angles between 180 and 270 lie in the third quadrant. The value of only Tan is positive in this quadrant. Shown as T (represents Tan) in the second diagram given below. 

 

Fourth Quadrant: All the angles between 270 and 360 lie in the fourth quadrant. The value of only Cos is positive in this quadrant. Shown as C (represents Cos) in the second diagram given below. 

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Starting from the fourth quadrant we can say that quadrant follows the series of CAST where C is for Cos, A for all, S  stands for Sin, and T is for Tan.

 

Related Angles

The value of sin, cos, and tan of some angles are equal to the values of sin, cos, and tan of other angles. Let us take an example of cos(-30⁰). Since 30 degrees lie in the first quadrant we can say that cos(-30⁰) is equal to cos(30⁰) because all the angles in the first quadrant are positive. Similarly, cos(390⁰) also equal to cos(30⁰). The diagrams below show the shaded angle of sin, cos, and tan having the same magnitude. 

Fig 1: sin 30 = 0.5

Fig 2: sin 150 = 0.5

Fig 3: sin 210 = -0.5

Fig 4: sin 330 = -0.5

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Therefore these angles are called related angles. The image given below shows the value of all angles of cos.

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Cos 120

The value of Cos 120 is - ½. It is the additive inverse of the value of cosine 60 or cos 60. Cosine is one of the basic trigonometric functions. It is expressed as a ratio of the base of a right-angled triangle to its hypotenuse. 


How to Find the Value of Cos 120?

The angles of a right-angled triangle are expressed in terms of multiples or sub-multiples of 180, or π in radians. Hence, to find the value of cos 120, we will have to express 120in terms of 180or 90


Case 1: 

Let us express 120 as (180 - 60)  

cos 120= cos (180 - 60)  

Since, cos (180- x) = - cos x.

Therefore, cos (180 - 60)= - cos 60 

=> cos 120= -½ 


Case 2: 

Let us express 120 as (90 + 30)

cos 120= cos (90 + 30)

Since, cos (90⁰ + x) = - sin x.

Therefore, cos (90 + 30) = -sin 30 

=>  cos 120= - ½ 


Solved Examples 


Question 1: Find the Exact Value of cos (- 390).

 

Solution:  We know cos(-x) = cos(x) 

therefore , cos (- 390) = cos (390). 

Since the angle, 390 is greater than angle 360

we find an angle t such that,

t = 390 - (360) = 30

This means Cos of cos (390) and cos ( 30)  coterminal.

So,

Cos (390) = Cos ( 30)

We know that the value of Cos 30 is equal to \[\sqrt{\frac{3}{2}}\]

FAQ (Frequently Asked Questions)

Question: Why do We Use sin theta and cos theta in an Angle?

Answer: Sin theta and Cos theta explains an interesting relation between the sides of the right-angled triangle. Pythagoras theorem can also be used to find the length of the missing side of a right-angled triangle if the other two sides are known.

Functions like Sin, Cos, and Tan are also used to find the side of a right-angled triangle if one side and the angle is known. For example, a right angle triangle has 60 degrees as theta and hypotenuse is 10 cm.

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We know, Cos = Adjacent /Hypotenuse

According to the table of values of trigonometric ratios, Cos (60) = ½ .

So, placing the value of Hypotenuse,

Cos 60 = Adjacent /10 cm

1/2 = Adjacent /10 cm

Adjacent = 10 / 2 = 5 cm.